Casselman subrepresentation theorem - Imported Knowlpedia Content

Statement

Let $G$ be a real reductive group, $K$ a maximal compact subgroup, and $P=MAN$ a minimal parabolic subgroup. The Casselman subrepresentation theorem states that every irreducible Harish–Chandra module $V$ for $(\mathfrak g,K)$ admits an injective $(\mathfrak g,K)$ -homomorphism

V\hookrightarrow \operatorname{Ind}_{P}^{G}(\sigma\otimes e^\nu\otimes 1_N)_{K\text{-finite}}

for some finite-dimensional irreducible representation $\sigma$ of $M$ and some $\nu\in\mathfrak a_{\mathbb C}^{*}$ . Thus every irreducible admissible module occurs as a submodule of a generally nonunitary principal series. The embedding is algebraic and need not split.

Meaning of the hypotheses

A Harish–Chandra module here is a finitely generated admissible $(\mathfrak g,K)$ -module on which the $K$ -action is algebraic and compatible with the differentiated $\mathfrak g$ -action. Irreducibility is taken in this algebraic category. The principal series in the target is likewise replaced by its $K$ -finite vectors, so the theorem does not assert an embedding between arbitrary Hilbert completions.

Proof mechanism and uses

Casselman’s Jacquet-module construction produces a nonzero exponent and finite-dimensional $MA$ -data. Frobenius reciprocity then converts the resulting quotient of a Jacquet module into an embedding of $V$ into induced representation data. This connects asymptotic expansions of matrix coefficients with principal series and underlies comparison results for globalizations Casselman, pp. 557–563.

Scope and contrast

The theorem is complementary to the Langlands quotient theorem: Casselman embeds an irreducible module into some principal series, whereas Langlands realizes it as the unique quotient of a positively ordered standard module. Neither statement says that every irreducible representation is itself a full principal-series module, and the inducing data in Casselman’s embedding need not be unique.

References

William Casselman, “Jacquet Modules for Real Reductive Groups,” in Proceedings of the International Congress of Mathematicians, Helsinki 1978, vol. 1, Academia Scientiarum Fennica, 1980, 557–563. IMU proceedings PDF. Relevant: Jacquet modules, asymptotics, and embeddings into induced representations.
William Casselman, “Jacquet Modules for Real Reductive Groups.” Author-maintained publication record. Relevant: bibliographic record and original article pagination.